Download Belief Regimes and Sovereign Debt Crises

Belief Regimes and Sovereign Debt Crises∗
Mark Aguiar Princeton University
Satyajit Chatterjee Federal Reserve Bank of Philadelphia
Harold Cole University of Pennsylvania
Zachary Stangebye University of Notre Dame
March 31, 2016
Preliminary
Abstract
Sovereign debt spreads occasionally exhibit sharp, large spikes in spreads over riskfree bonds. We document that these movements are only weakly correlated with movements in domestic output and are frequently followed by reductions in the face value of
debt outstanding. Motivated by this evidence, we propose a quantitative model with
long-term bonds and three sources of risk: fluctuations in the growth of domestic income; movements in the risk premia associated with default risk; and shifts in creditor
“beliefs” regarding the actions of other creditors. We show that the shifts in creditor
beliefs directly play an important role in generating default risk, but also amplify the
impact of shocks to fundamentals. Interestingly, persistent changes to risk premia have
a negligible impact on spreads, and an increase in risk premium may even lead to a
decline in spreads. The latter reflects that a higher risk premium provides discipline
regarding future debt issuances. More generally, the sovereign borrowing decisions are
quantitatively sensitive to equilibrium bond prices. Even large, relatively unexpected
shocks to creditor beliefs have only a modest effect on spreads as the government
responds by aggressively deleveraging.
∗
The views expressed here are those of the authors and do not necessarily represent the views of the Federal
Reserve Bank of Philadelphia or the Federal Reserve System. We thank Manuel Amador for numerous helpful
comments. We also thank our discussant Luigi Paciello.
1
1
Introduction
In this paper, we study an environment in which sovereign debtors face risk due to shocks
to the endowment, shocks to the risk premia charged in loan markets, and shocks to creditor “beliefs” regarding the actions of other creditors. We motivate this environment by
documenting a number of facts regarding emerging market bonds. First, as is well known,
emerging market spreads over benchmark risk-free bonds are volatile. Second, while large
spikes in spreads are correlated with declines in output, the correlation is relatively weak. In
fact, a sizable proportion of such spikes occur when growth is positive and in line with historical means. Third, we show that while some of these unexplained movements in spreads
can be explained by shifts in measures of global risk premia, there remains a large and timevarying residual component to movements in spreads. One possible interpretation of this
residual source of risk is that the likelihood of a self-fulfilling debt crisis is time varying.
It is this latter notion that we embed in our quantitative model to study the associated
equilibrium outcomes.
Specifically, we consider a small open economy that trades a non-contingent (but defaultable) bond with a pool of creditors and a constant world risk-free interest rate, as
in Eaton and Gersovitz (1981) and the recent quantitative literature starting with Aguiar
and Gopinath (2006) and Arellano (2008). The economy’s endowment process is subject
to shocks to both transitory and trend output, as in (Aguiar and Gopinath, 2006, 2007).
The creditors involved in sovereign lending are risk-averse with finite wealth, and hence the
sovereign pays a risk premium. We allow the pool of creditors, and the associated stock
of wealth to be invested, to vary over time exogenously, generating another source of risk.
Finally, we build on the timing introduced by Cole and Kehoe (2000) to support multiple
Markov equilibria which may involve rollover crises. We consider an equilibrium such that
creditor beliefs regarding the probability of a rollover crisis is time varying. To the best of
our knowledge, this is the first quantitative model that incorporates all three sources of risk.1
The process for creditor beliefs follows a three-state Markov process. One belief regime
is a rollover crisis a la Cole and Kehoe (2000), in which the government is forced to default
by a run on its debt. A second belief regime is a “tranquil” regime, in which the probability
of a run in the next quarter is negligible. The middle regime is a “vulnerable” regime, in
which creditors do not run in the current quarter, but consider the likelihood of a run in the
1
A related recent effort is Bocola and Dovis 2015, which attempts to decompose the relative contribution
of each factor in the recent crisis in Italy.
2
next quarter to be relatively high.
We calibrate the model to Mexico and explore the role of the multiple sources of risk. We
find that roughly half of the simulated debt crises, defined as a jump in spread in the upper
5th percentile of the distribution, occur when creditors believe a rollover crisis is imminent.
The other half occur when creditor beliefs are relatively “tranquil,” despite the fact that
we calibrate beliefs to be tranquil for over 80 percent of the periods. Debt crises in the
tranquil-belief regime are associated with boom-bust cycles in output. The preceding boom
is necessary to generate large debt positions, which makes default probabilities sensitive to
the subsequent bust. Shifts in creditor beliefs can generate sharp spikes in spreads in the
absence of declines in fundamentals, bringing us closer to the patterns documented in the
data.
Interestingly, shocks to creditor wealth are relatively unimportant in generating large
spikes in spreads. For high levels of debt, a drop in creditor wealth raises the risk premium
charged on sovereign bonds and generates an increase in spreads. However, for moderate
levels of debt, spreads may actually fall with a drop in creditor wealth. This is because
wealth shocks are persistent and provide a disincentive to issue more debt in the future,
mitigating the dilution problem associated with long-maturity bonds.
A switch from tranquil to vulnerable beliefs generates a sharp increase in spreads. The
sovereign either immediately defaults, or begins to reduce debt quickly. Along the path,
it may be hit with a rollover crisis, but conditional on survival it reduces its level of debt.
These dynamics are reminiscent of the theoretical insights of Cole and Kehoe (2000).
We calibrate the model so that defaults occur on average twice every hundred years.
Only about half of the outright defaults in the simulation occur when creditor beliefs are
in the tranquil regime, despite the fact that this regime accounts for the vast majority of
the periods. These are triggered by large declines in endowment growth, as in the standard
Eaton-Gersovitz framework. Approximately 10 percent of the defaults occur in the vulnerable regime and another 40 percent are due to outright rollover crises. The defaults when
beliefs are tranquil are not unrelated to the probability of a rollover crisis at some point in the
future. We compute an alternative model in which rollover crises never occur and evaluate
default decisions using these alternative policy functions. All of the simulated defaults in the
benchmark model occur in parts of the state space in which the sovereign would repay debt
if rollover crises are always probability zero events. This reflects that the tradeoff between
defaulting versus retaining access to credit markets is fundamentally changed when creditor
3
beliefs are time varying.
2
Motivating Facts
2.1
Data for Emerging Markets
We start with a set of facts that guides our model of sovereign debt crises. Our sample
spans the period 1993Q4 through 2014Q4, and includes data from 20 emerging markets:
Argentina, Brazil, Bulgaria, Chile, Columbia, Hungary, India, Indonesia, Latvia, Lithuania,
Malaysia, Mexico, Peru, Philippines, Poland, Romania, Russia, South Africa, Turkey, and
Ukraine. For each of these economies, we have data on GDP in US dollars measured in 2005
domestic prices and exchange rates (real GDP), GDP in US dollars measured in current
prices and exchange rates (nominal GDP), gross external debt in US dollars (debt), and on
market spreads on sovereign debt.2
Tables 1 and 2 report summary statistics for the sample.3 Table 1 document the high and
volatile spreads that characterized emerging market sovereign bonds during this period. The
standard deviation of the level and quarterly change in spreads are 6.76 and 2.29 percentage
points, respectively. Table 2 reports an average external debt-to-(annualized)GDP ratio of
0.46. This level is low relative to the public debt levels observed in developed economies.
The fact that emerging markets generate high spreads at relatively low levels of debt-to-GDP
reflects one aspect of the “debt intolerance” of these economies documented by Reinhart,
Rogoff, and Savastano (2003).
While many of the countries in our sample have very high spreads, only two - Russia
in 1998 and Argentina in 2001 - ended up defaulting on their external debt, while a third,
Ukraine, defaulted on its internal debt (in 1998). This reflects that defaults are relatively
rare events; Tomz and Wright (2013) document over a larger sample that default occurs with
an unconditional probability of roughly 2 percent.
2
Data source for GDP and debt is Haver Analytics’ Emerge database. The source of the spread data is
JP Morgan’s Emerging Market Bond Index (EMBI).
3
Note that Russia defaulted in 1998 and Argentina in 2001, and while secondary market spreads continued
to be recorded post default, these do not shed light on the cost of new borrowing as the governments were
shut out of international bond markets until they reached a settlement with creditors. Similarly, the face
value of debt is carried throughout the default period for these economies.
4
Table 1: Sovereign Spreads: Summary Statistics
Country
Mean
r − r?
Std Dev Std Dev
r − r?
∆(r − r? )
Argentina
Brazil
Bulgaria
Chile
Columbia
Hungary
India
Indonesia
Latvia
Lithuania
Malaysia
Mexico
Peru
Philippines
Poland
Romania
Russia
South Africa
Turkey
Ukraine
15.25
5.60
5.24
1.46
3.48
1.82
2.25
2.85
1.57
2.46
1.75
3.45
3.43
3.43
1.91
2.71
7.10
2.26
3.95
7.60
17.59
3.93
4.86
0.57
2.06
1.54
0.54
1.37
0.34
0.92
1.22
2.53
1.96
1.53
1.38
1.02
10.96
1.16
2.17
6.07
6.10
1.74
1.29
0.34
0.88
0.57
0.47
0.98
0.16
0.48
0.75
1.34
0.84
0.75
0.54
0.49
4.78
0.68
0.95
3.50
7.17
2.04
1.55
0.34
2.45
0.88
0.85
0.73
0.17
0.98
0.81
1.27
1.82
1.36
0.67
0.68
1.75
0.99
2.05
5.77
0.18
0.09
0.03
0.00
0.05
0.02
0.00
0.02
0.00
0.00
0.03
0.05
0.06
0.04
0.01
0.00
0.06
0.03
0.05
0.11
Pooled
4.31
6.76
2.29
1.58
0.05
5
95th pct Frequency
∆(r − r? )
Crisis
Table 2: Sovereign Spreads: Summary Statistics
Mean
B
4∗Y
Corr
(∆(r − r? ), ∆y)
Corr
(r − r? , %∆B)
Corr
(∆(r − r? ), %∆B)
Argentina
Brazil
Bulgaria
Chile
Columbia
Hungary
India
Indonesia
Latvia
Lithuania
Malaysia
Mexico
Peru
Philippines
Poland
Romania
Russia
South
Turkey
Ukraine
0.38
0.25
0.77
0.41
0.27
0.77
0.82
0.18
0.49
1.06
0.54
0.16
0.48
0.47
0.57
0.61
NA
0.26
0.38
0.64
-0.35
-0.11
0.09
-0.16
-0.29
-0.24
-0.32
-0.43
-0.18
-0.25
-0.56
-0.4
-0.01
-0.16
-0.09
0.5
-0.45
-0.14
-0.34
-0.49
-0.22
-0.18
-0.20
-0.18
-0.40
-0.56
0.04
-0.03
-0.12
-0.17
-0.33
0.23
-0.39
0.06
-0.35
0.42
-0.30
-0.38
-0.20
-0.60
0.08
-0.01
0.06
-0.11
-0.07
-0.05
-0.65
0.07
-0.16
-0.31
0.24
-0.13
-0.05
0.09
-0.38
-0.33
0.02
-0.24
0.08
-0.07
Pooled
0.46
-0.27
-0.19
0.01
Country
6
2.2
Debt Crises
One of the first facts that we want to document is that there is a relatively weak link between
a country’s fundamentals and whether or not it has a debt crises. We consider a debt crisis
to be a large spike in spreads, regardless of whether the country defaults or not. From Table
1, we see that the 95th percentile of the quarterly change in spread is 158 basis points.
We define a debt crisis to be a quarterly change in spreads that exceeds this threshold.
By definition, 5 percent of the sample involves a debt crisis. However, this is not equally
distributed across countries. From Table 1, we see that five countries never have experience
a crisis, while nearly 20 percent of Argentina’s quarterly observations are above the crisis
threshold.
An important question is whether debt crises are associated with domestic fundamentals.
Table 2 reports a generally negative correlation between the change in spreads and output.
However, this statistic masks surprising heterogeneity across crisis episodes. Figure 1 depicts
the density of GDP growth in crisis and non-crisis quarters. Specifically, each quarter is
assigned to a crisis or non-crisis category, depending on whether the change in EMBI spread
from last quarter exceeds the 95th percentile threshold. We then plot the estimated kernel
density of quarterly GDP growth for each subsample. In Panel (a), the solid line depicts
the kernel density of quarterly growth in real GDP in quarters defined as non-crisis, while
the dashed line depicts the corresponding histogram for crisis quarters. Panel (b) redefines
growth as the average growth over the preceding four quarters; that is, if period t is used to
define crisis status (that is, rt − rt−1 > 1.58), then growth is averaged over quarters t − 5 and
t − 1 (that is, (lnYt−1 − ln Yt−5 )/4). Panel (c) looks at subsequent growth; that is, average
growth between period t and t + 4.
By comparing the two densities in Panel (a), one can see from the increase in the frequency of negative growth rates for countries experiencing crises relative to the overall distribution that there is a higher tendency for negative growth rates to be associated with crises.
However, what is striking is the extent to which countries experiencing positive growth also
experienced crises. Overall, the graph indicates some association between negative growth
and debt crises, but the association is not strong. Specifically, while mean contemporaneous
(quarterly) growth during a crisis is -1.2, as opposed to 1.0 during non-crisis quarters, nearly
half (46 percent) of the crisis periods have strictly positive growth.
Not only do contemporaneous fundamentals have a hard time accounting for debt crises,
they do an even worse job of forecasting debt crises. Specifically, in Panel (b) we see that
7
Figure 1: Sovereign Debt Crises and Growth
(b) Preceding-Year Average Growth
0
0
10
20
Density
Density
20
30
40
40
60
50
(a) Contemporaneous Growth
-.1
-.05
0
Growth
Crisis
.05
-.05
0
Growth
No Crisis
Crisis
.05
No Crisis
0
20
Density
40
60
(c) Subsequent-Year Average Growth
-.05
0
Growth
Crisis
.05
No Crisis
Note: Each panel overlays two histograms (kernel densities) of GDP growth. The sample consists of 23 emerging markets between 1993Q4 and
2014Q4 (see text). The histogram labelled “No Crisis” refers to periods in which the quarterly change in sovereign bond spreads is less than 158
basis points. The histogram labelled “Crisis” refers to periods in which the change in spread is greater than or equal to 158 basis points. This
threshold is chosen so that 5 percent of the periods are defined as “Crisis.” In Panel (a), growth is defined as the quarterly change in log GDP
contemporaneous with the quarterly chang in spreads used to define a crisis. Specifically, if a crisis occurs in quarter t due to rt − rt−1 > 185,
where rt is the spread over the risk-free rate observed in quarter t, then growth is yt − yt−1 , where yt is log GDP in quarter t. In Panel (b),
growth is averaged over the year preceding the quarter used to define a crisis; that is (yt−1 − yt−5 )/4. In Panel (c), growth is averaged over the
subsequent year; that is (yt+4 − yt )/4.
8
growth is often quite positive in the year prior to the jump in spreads. Here the association
between growth and crises is substantially weaker than with contemporaneous growth rates.
The lack of contemporaneous and lagged association between growth debt crises leaves
open the possibility that it is the anticipation of future bad fundamentals that lead to a rise
in spreads and debt crises. If such news shocks played an important role, then one would
expect to see high spreads forecast negative future fundamentals. In the third panel we
illustrate the lack of a tight connection between news about fundamentals and spreads by
again graphing growth rate histograms, but now the density is for growth rates in the year
following a debt crises. The number of crises followed by positive growth is remarkable given
that there are many reasons to believe high spreads should suppress economic activity (see,
for example, Neumeyer and Perri, 2005).
This evidence suggests that while negative GDP growth is correlated with increases in
spreads, the association is rather weak. This fact complements the finding of (Tomz and
Wright, 2007) that a significant fraction (roughly 40%) of defaults occur when GDP is above
trend. While shocks to output are a natural starting point for understanding sovereign debt
crises, the data suggest there is much more to the story.
2.3
Deleveraging
The data from emerging markets can also shed light on debt dynamics during a crisis. Table
2 documents that periods of above-average spreads are associated with reductions in the face
value of gross external debt. The pooled correlation of spreads at time t and the percentage
change in debt between t−1 and t is −0.19. The correlation of the change in spread and debt
is roughly zero. However, a large change in spread (that is, a crisis period) is associated with
a subsequent decline in debt. In particular, regressing the percent change in debt between t
and t + 1 on the indicator for a crisis in period t generates a coefficient of -1.6 and a t-stat
of nearly 3. This relationship is robust to the inclusion of country fixed effects. This implies
that a sharp spike in spreads is associated with a subsequent decline in the face value of
debt. Note that the decline in the level of outstanding debt is not always associated with a
decline in the debt-to-GDP ratio. From Figure 1 we know that many crises are associated
with declines in GDP, which frequently are larger than the percentage declines in debt.
9
2.4
Excess Returns
Another possible source of volatility in sovereign debt spreads is shifts in the appetite for
risk. Sovereign bonds appear to earn an excess return, and the extent of this return is
correlated with such global risk factors as US stock returns and volatility indices (Pan and
Singleton, 2008; Longstaff, Pan, Pedersen, and Singleton, 2011; Borri and Verdelhan, 2011).
A similar pattern emerges from our sample of emerging markets. In particular, we compute
the realized returns on a value-weighted portfolio of actively traded, emerging country bonds,
specifically the EMBI+ index constructed by JP Morgan. Table 3 reports the return for the
full sample period the index is available, as well as two sub-periods. The table also reports
the returns to portfolio of 2-year and 5-year maturity U.S. Treasury debt. We offer two
risk-free references as the EMBI+ does not have a fixed maturity structure and typically
ranges between 2 and 5 years.
The EMBI+ index paid a return in excess of the risk-free portfolio of five to six percent.
This excess return is roughly stable across our the two sub-periods as well. The index
reported in Table 3 represents a fairly diversified portfolio of emerging economy debt. The
underlying country-level returns range from 1.3 percent for Argentina to over 10 percent
for Brazil, Bulgaria, and Peru. Whether the realized return reflects the ex ante expected
return depends on whether our sample accurately reflects the population distribution of
default and repayment. The assumption is that by pooling a portfolio of bonds, the EBMI+
followed over 20 years of data provides a fair indication of the expected return on a typical
emerging market bond. Of course, we cannot rule out the possibility that this sample is not
representative. Nevertheless, the observed returns are consistent with a fairly substantial
risk premia charged to sovereign borrowers.
We explore the impact of global risk factors on country-level spreads in Table 4. In the
first two columns of panel (a), we report the results of regressing the level of spreads (at
the close of each quarter) on the contemporaneous level of the P/E ratio or the VIX as well
as country fixed effects. The last two columns perform the same regressions with the level
of the log bond price index as the dependent variable. Panel (b) reports the results of the
same regressions using quarterly first differences. For the price index, we compute quarterly
log changes; for all other variables the differences are in levels. In levels, an increase in the
P/E ratio, which implies a decrease in the associated risk premium, is associated with an
increase in spreads, and a decline in bond prices. However, this pattern is reversed if the
VIX is used as the measure of risk.
10
Table 3: Realized Bond Returns
Period
1993Q1–2014Q4
1993Q1–2003Q4
2004Q1–2014Q4
EMBI+
2-Year
Treasury
5-Year
Treasury
9.7
11.1
8.2
3.7
5.4
2.0
4.7
6.3
3.1
Panel (b) suggests that in first differences, an increase in risk raises spreads and lowers
prices, regardless of which measure we use as a proxy for global appetite for risk. Contrasting
the two panels suggests that emerging market spreads co-move with global risk premia at high
frequencies, but this relationship may be weaker or reversed at lower frequencies. Given the
relatively short time frame of the exercise, the latter statement is speculative. Nevertheless,
we will see in the model that a low frequency reduction in creditors’ appetite for risk may
indeed generate higher prices and lower spreads.
2.5
The European Crisis
The above facts concerned emerging markets, which have generated most of the post-war
debt crises. However, the recent crisis in Europe has renewed interest in debt crises in more
advanced economies. [TBA]
2.6
Taking Stock
The empirical facts documented above suggest sovereign debt crises are more than just
the result of poor fundamentals, which is the natural starting point of the quickly growing
literature on sovereign default. A successful quantitative model should address the following
empirical regularities:
1. Crises, and particularly defaults, are low probability events;
2. Crises are not tightly connected to poor fundamentals;
3. Spreads are highly volatile;
11
Table 4: Global Risk Factors
(a) Levels
rt − rt∗
P/E Ratio
26.24
(10.13)
VIX
R2
N
Countries
rt − rt∗
ln pt
-0.052
(0.007)
14.81
(3.41)
0.28
1,250
22
ln pt
0.29
1,250
22
-0.007
(0.002)
0.41
1,282
22
0.31
1,282
22
(b) Differences
∆(rt − rt∗ )
∆ P/E Ratio
∆(rt − rt∗ )
-30.06
(5.05)
∆ ln pt
0.011
(0.002)
∆ VIX
R2
N
Countries
∆ ln pt
10.68
(2.04)
0.04
1,228
22
0.08
1,228
22
12
-0.0036
(0.0008)
0.05
1,260
22
0.08
1,260
22
4. Rising spreads are associated with de-leveraging by the sovereign; and
5. Risk premia are an important component of sovereign spreads.
In considering what features of real-world economies are important in generating these
patterns, the first thing to recognize is that sovereign debt lacks a direct enforcement mechanism: most countries default despite having the physical capacity to repay. Yet, countries
seem perfectly willing to service significant amounts of debts most of the time (rescheduling
of debts and outright default are relatively rare events). Without any deadweight costs of
default, the level of debt that a sovereign would be willing to repay is constrained by the
worst punishment lenders can inflict on the sovereign, namely, permanent exclusion from all
forms of future credit. It is well known that this punishment is generally too weak, quantitatively speaking, to sustain much debt (this is spelled out in a numerical example in Aguiar
and Gopinath, 2006). Thus, we need to posit deadweight costs of default.
Second, defaults actually occurring in equilibrium reflect that debt contracts are not
fully state-contingent, and default provides an implicit form of insurance. However, even
with rational risk-neutral lenders who break-even on average for every loan extended to
sovereigns, the deadweight cost of default (which do not accrue to lenders) makes default
an actuarially unfair form of insurance against bad states of the world. This insurancethrough-default becomes even more actuarially unfair with risk-averse lenders. Moreover,
sovereigns can self-insure by accumulating reserves as precautionary savings, which is an
attractive option for smoothing transitory shocks. Given that this suggests fairly substantial
deadweight costs of default and substantial risk aversion on the part of lenders, the insurance
offered by the possibility of default appears to be quite costly in practice. The fact that
countries carry large external debt positions despite the costs suggests that sovereigns are
fairly impatient, perhaps reflecting political economy frictions.
However, while myopia can explain in part why sovereigns borrow, it does not necessarily
explain why they default. As noted already, default is a very costly form of insurance against
bad states of the world. This fact – via equilibrium prices – can be expected to encourage the
sovereign to stay away from debt levels for which the probability of default is significant. This
has two implications. First, when crises/defaults do materialize, they come as a surprise,
which is consistent with these events being low probability events. Unfortunately, the other
side of this coin is that getting the mean and volatility of spreads right is a challenge for
quantitative models, especially once we incorporate that many shifts in spreads occur in
13
relatively good economic times. Getting high and variable spreads means getting periods of
high default risk as well as substantial variation in expected future default risk. This will
be difficult to achieve when the borrower has a strong incentive to adjust its debt-to-output
level to the point where the probability of future default is (uniformly) low.
This discussion highlights the fact that getting both high and time-varying spreads, high
debt-to-output and infrequent crisis and defaults poses a challenge. One way to attack the
problem is to leverage the fact that an important element of sovereign borrowing is the
sovereign’s need to roll over its debts (the time horizon of investors is typically shorter than
the time horizon of the sovereign). The inability to rollover one’s debt can induce default in
circumstance in which the borrower would have repaid with the rollover. From the lender’s
perspective a rollover default represents a coordination failure if repayment would have
occurred with rollover. This channel has been thought to be important for explaining the
sorts of debt crises we see (e.g. Calvo, 1988; Cole and Kehoe, 2000). A contribution of the
current study is to quantify how the possibility of rollover default impacts the level and the
time variation in spreads, and whether that matches the data.
3
Environment
We consider a single-good, discrete-time environment, with time indexed by t = 0, 1, ....
The focus of the analysis is a small open economy that receives a stochastic endowment.
The economy is small relative to the rest of the world in the sense that its endowment
realizations and decisions do not affect the world risk-free interest rate. However, financial
markets are segmented in the sense that the economy can borrow from a set of potential
lenders with limited wealth. Consumption and saving decisions are made on behalf of the
domestic economy by a sovereign government. In this section, we proceed by characterizing
the domestic economy and the sovereign’s problem, then turn to the lenders’ problem, and
conclude by defining an equilibrium in the sovereign debt market.
14
3.1
3.1.1
The Domestic Economy
Technology
The economy receives a stochastic endowment Yt > 0 each period. The endowment process
is characterized by:
Yt = Gt ezt ,
(1)
where
ln Gt ≡
t
X
gs ,
(2)
s=1
is the cumulation of period growth rates gt , and zt represents fluctuations around trend
growth. We assume that gt and zt follow finite-state first-order Markov processes. The
relevant state vector for the current endowment and its probability distribution going forward
is (Yt , gt , zt ).
3.1.2
Preferences
The domestic economy is run by an infinitely lived sovereign government, which enjoys
preferences over the sequence of aggregate consumption {Ct }∞
t=0 given by:
E
∞
X
β t u(Ct ),
t=0
with β ∈ (0, 1) and
C 1−σ
,
1−σ
with σ 6= 1. We assume that the sovereign has enough instruments to implement any feasible
consumption sequence as a domestic competitive equilibrium, and therefore abstract from
the problem of individual residents of the domestic economy. This does not mean that the
government necessarily shares the preferences of its constituents, but rather that it is the
relevant decision maker viz a viz international financial markets.
u(C) =
To ensure that the government’s problem is well behaved, we require:
n
o
0
max Eg βe(1−σ)g < 1,
g
15
where the max is taken over elements of the Markov process for gt .
3.1.3
Financial Markets
The sovereign issues non-contingent bonds to a competitive pool of lenders (described below).
Bonds pay a coupon every period up to and including the period of maturity, which, without
loss of generality, we normalize to r∗ per unit of face value, where r∗ is the (constant)
international risk-free rate. With this normalization, a risk-free bond will have an equilibrium
price of one. For tractability, we consider a bond with random maturity, as in Leland (1994).4
In particular, each bond matures next period with a constant hazard rate λ ∈ [0, 1]. We
let the unit of a bond be infinitesimally small, and let maturity be iid across individual
bonds, such that with probability one a fraction λ of any non-degenerate portfolio of bonds
matures each period. The constant hazard of maturity implies that all bonds are symmetric
before the realization of maturity at the start of the period, regardless of when they were
purchased. Note as well the expected maturity of a bond is 1/λ periods, and so λ = 0
is a console and λ = 1 is one-period debt. While we vary λ across quantitative exercises,
within any specific environment there is only one maturity traded. With these conventions,
a portfolio of sovereign bonds of measure x receives a payment (absent default) of (r∗ + λ)x,
and has a continuation face value of (1 − λ)x.
We denote the outstanding stock of debt at the start of period t by Bt . We do not restrict
the sign of Bt , which allows the government to be either a net creditor (B < 0) or debtor
(B > 0). Net issuances of new debt in period t is given by Bt+1 − (1 − λ)Bt , where (1 − λ)
is the fraction of debt that does not mature in the current period. If Bt+1 < (1 − λ)Bt ,
then the government is repurchasing its outstanding debt rather than issuing new debt. We
denote the debt-to-income ratio by bt ≡ BYtt . To rule out Ponzi schemes, we place an upper
bound on the debt-to-income ratio: bt ≤ b, ∀t.
3.1.4
Timing
The timing of a period is depicted in Figure 2. Let s denote the aggregate state after the period’s realization of random variables but before the period’s consumption and debt-issuance
decisions have been made. Specifically, s = (Y, g, z, b, w, ρ), where (Y, g, z) characterize the
4
See also Hatchondo and Martinez (2009), Chatterjee and Eyigungor (2012) and Arellano and Ramanarayanan (2012).
16
current period endowment state; b is the inherited stock of debt normalized by Y ; w is the
stock of lender wealth also normalized by Y and discussed in Section 3.3; and ρ is a random
variable that coordinates equilibrium beliefs and is discussed in Section 3.5. As Y is an
element of s, the normalization of the other variables is without loss of generality. Let S
denote the set of possible states s. Other than b, the elements of s follow an exogenous
Markov process.
No Default
Initial
State: s
Auction
B 0 − (1 − λ)B
at price
q(s, b0 )
C
Y
= 1 − (r∗ + λ)b +
q(b0 − (1 − λ)b)
Next
Period: s0
Settlement
V D (s)
Default
Figure 2: Timing within a Period
After observing the period’s realized s, the government decides to auction B 0 − (1 − λ)B
units of debt, where B 0 represents the face value of debt at the start of the next period. There
is one auction per period. While this assumption is standard, it does allow the government
to commit to the amount auctioned within a period.5 Let b0 denote next period’s face value
of debt normalized by the current endowment: b0t ≡ BYt+1
. The evolution of debt-to-income
t
is then:
Yt
= b0t e−gt+1 −zt+1 +zt .
bt+1 = b0t
Yt+1
We postpone the formal definition of equilibrium until Section 3.4, but to anticipate we
shall consider equilibria in which endogenous variables are functions of the state s ∈ S.
Let q(s, b0 ) denote the equilibrium price schedule. The government is large in its own debt
market and internalizes the fact that it faces different prices depending on how much debt
it auctions; hence in choosing, b0 the government internalizes the entire price schedule as
a function of b0 . Given outstanding debt B, the government is contractually obligated to
pay λB in principal and r∗ B in interest payments. These payments are financed through
5
For an exploration of an environment in which the government cannot commit to a single auction, see
Lorenzoni and Werning (2013).
17
current endowment as well as new debt issuances, and are made in a sub-period labeled
“settlement” in figure 2. If the government makes its contracted payments, it consumes
C = Y [1 − (r∗ + λ)b + q(s, b0 )(b0 − (1 − λ)b)], and continues on to the next period with the
new debt state governed by b0 .
If the government fails to make the contracted payment, the government enters a default
state the next period. While in default, the government has no access to foreign financial
markets. Moreover, in the default state the government loses part of its endowment, which
proxies for economic disruptions that are a consequence of default in practice. Let φ(g, z)
denote the proportional loss of output, so that in the default state the government receives
(1 − φ(g, z))Y when the non-default state endowment is Y . The proportion lost is a function
of the realized endowment shocks (g, z), but is otherwise independent of the level of income
Y . While in default, the government has a constant hazard ξ ∈ [0, 1] of exiting the default
state. Having exited default, the government no longer suffers an endowment penalty and
regains access to foreign financial markets. The debt outstanding at the time of default is
forgiven, implying that creditors are paid zero in a default.
If the government defaults at settlement, it does not receive the proceeds from that
period’s auction. In particular, it consumes its endowment (minus the cost of default). The
amount raised via auction, if any, is disbursed to outstanding bondholders in proportion
to the face value of their bond positions. An important implication of this convention is
that holders of newly issued bonds are not fully compensated if the government defaults
immediately after the auction. That is, each unit of debt is treated equally and receives
q(s, b0 )(B 0 − (1 − λ)B)/B 0 , which is strictly less than the price paid as long as B > 0, and
therefore implies an immediate loss for new bondholders. In this regard, our timing deviates
from that of Eaton and Gersovitz (1981), which has become standard in the quantitative
sovereign debt literature. In the standard timing, the bond auction occurs after that period’s
default decision has been made. Thus newly auctioned bonds do not face within-period
default risk. Our timing expands the set of equilibria relative to the Eaton-Gersovitz timing,
and in particular allows a tractable way of introducing self-fulfilling debt crises.6
6
The timing in Figure 2 is adapted from Aguiar and Amador (2014b), which in turn is a modification of
Cole and Kehoe (2000). The difference relative to Cole and Kehoe is that we do not allow the government
to consume the proceeds of an auction if it defaults. This simplifies the off-equilibrium analysis without
materially changing the results. See Auclert and Rognlie (2014) for a discussion of how the Eaton-Gersovitz
timing in some standard environments has a unique Markov equilibrium, thus ruling out self-fulfilling crises.
18
3.2
The Government’s Problem
Let V (s) denote the start-of-period value for the government, conditional on the state s and
the equilibrium price schedule q (which we suppress in the notation). Working backwards
through a period, at the time of settlement the government has issued B 0 − (1 − λ)B units
of new debt at price q(s, b0 ) and owes (r∗ + λ)B.
If the government honors its obligations at settlement, its payoff is:
V R (s, b0 ) = u(C) + βE [V (s0 )|s, b0 ] ,
(3)
with
C = Y − (r∗ + λ)B + q(s, b0 )(B 0∗ − (1 − λ)B
= Y [1 − (r∗ + λ)b + q(s, b0 )(b0∗ − (1 − λ)b] .
Note that consumption is pinned down at settlement by the budget constraint; if the required
consumption is negative, we define V R (s, b0 ) = −∞, which is always dominated by default.
If the government defaults at settlement, its payoff is:
V D (s) = u(C) + β(1 − ξ)E V D (s0 )|s + βξE [V (s0 )|s, b0 = 0] ,
(4)
with
C = (1 − φ(s))Y,
and where we recall that ξ is the probability of exiting the default state. The output cost
of default is governed by the function φ. Note that s0 includes b0 as an element, and b0 = 0
while in the default state by definition. The amount of new debt implied by b0 is not relevant
for the default payoff as the government does not receive the auction proceeds if it defaults
at settlement.
The start-of-period value function is:
0
R
D
V (s) = max max V (s, b (s)), V (s) , ∀s ∈ S.
b0 ≤b̄
(5)
Let B : S → (−∞, b̄] denote the policy function for b0 generated by the government’s
19
problem. We denote the policy function for default at settlement conditional on b0 by D(s, b0 ).
For technical reasons, we allow the government to randomize over default and repayment
when indifferent; that is, when V R (s, b0 ) = V D (s). Therefore, D : S × −∞, b → [0, 1] is the
probability the government defaults at settlement, conditional on (s, b0 ). The policy function
of consumption is implied by those for debt and default.
3.3
Lenders
We assume financial markets are segmented and only a subset of foreign agents participate in
the sovereign debt market. This assumption allows us to introduce risk premia on sovereign
bonds as well as explore how shocks to foreign lenders’ wealth influence the domestic economy; all the while treating the risk-free rate as parametric. For tractability, we assume that
a set of lenders has access to the sovereign bond market for one period, and then exits,
to be replaced by a new set of lenders. The short horizon of the specialist lenders is for
tractability, avoiding the need to solve an infinite horizon portfolio problem and carry another endogenous state variable. Nevertheless, it allows us to capture the possibility that
exogenous shocks to lenders’ wealth affect the price of sovereign bonds.7
Specifically, each period a unit measure of identical lenders enter the sovereign debt
market. Let Wt denote the aggregate wealth of the agents that can participate in period t’s
bond market, and Wi,t represent the wealth of a (representative) individual agent i ∈ [0, 1].
Lenders allocate their wealth across sovereign bonds and a risk free asset that yields 1 + r∗ .
As noted above, the risk-free rate is pinned down by the larger world financial market, and
specialists in the sovereign bond market can freely borrow and lend at this rate.
The one-period return on sovereign bonds depends on the default decision within the
period as well as next period’s default decision. Let δt and δt+1 denote the government’s
realized default decisions in period t and at period t+1’s settlement, respectively. The return
also depends on period t + 1’s bond prices, as only a fraction λ of the bond portfolio matures
next period. Let qt and qt+1 denote the equilibrium price in t and t+1, respectively. A lender
which invests a fraction µ of wealth in sovereign bonds at price qt has an end-of-auction bond
7
An alternative is to specify an exogenous process for the stochastic discount factor. We view our approach
and this alternative as two ways to capture the same phenomenon.
20
position
µWi,t
,
q
and a next period wealth given by:
0
Wi,t
= (1 − µ)Wi,t (1 + r∗ ) + µ
Wi,t
(1 − δt )(1 − δt+1 )[r∗ + λ + (1 − λ)qt+1 ].
qt
(6)
0
Note that Wi,t
is used to denote the wealth of a period-t lender in t + 1; as Wi,t+1 represents
the wealth of a new entrant in t + 1.8
The representative lender’s decision is how much sovereign debt to purchase at auction
conditional on an equilibrium price and end-of-period debt b0 . Dropping time subscripts, a
representative lender i with individual wealth Wi at the start of a period solves the following
problem:
0
0
0 L(Wi , s, b ) = max E v (Wi ) s, b ,
µ
subject to (6), where
W 1−γ
,
v(W ) =
1−γ
with γ 6= 1. In forming expectations over Wi0 , the lender uses the equilibrium policy functions
of the government:
δt = 1 with probability D(st , b0t )
δt+1 = 1 with probability D(st+1 , B(st+1 )) in state st+1
qt+1 = q(st+1 , B(st+1 )).
This problem implies an optimal amount of debt purchased, conditional on (Wi , s, b0 ) and
an equilibrium price schedule q. Evaluating this policy at Wi = W , let L (s, b0 ) denote the
aggregate demand for sovereign debt normalized by Y ∈ s. Market clearing for sovereign
bonds is therefore:
L (s, b0 ) = b0 .
8
(7)
Note that in (6) we omit possible proceeds from an auction followed immediately by default. Recall
from the timing depicted in Figure 2 that in such an event, the proceeds are rebated proportionally to all
(legacy and new) bondholders. As this amount is always zero along the equilibrium path, we omit it from
(6) to minimize clutter.
21
The FOC for L (s, b0 ) implies:
q(s, b0 ) =
E[w0−γ (1 − δ(s, b0 ))(1 − δ(s0 , b0 )(r∗ + λ + (1 − λ)q(s0 , b00 (s0 )))]
,
(1 + r∗ )E[w0−γ ]
(8)
where w0 is representative lender’s next period wealth normalized by Y and b00 (s0 ) represents
the government’s choice of debt next period conditional on s0 . Equation (8) encompasses
well known special cases: If lenders are risk neutral and debt is short term (γ = 0 and λ = 1),
q(y, b0 ) is simply the probability of repayment on the debt; if lenders are risk neutral but the
debt is long-term (γ = 0 and λ > 0)
q(s, b0 ) =
3.4
E(1 − δ(s, b0 ))(1 − δ(s0 , b0 ))(r∗ + λ + (1 − λ)q(s0 , b00 (s0 )))]
.
(1 + r∗ )
(9)
Definition of Equilibrium
Definition 1 (Equilibrium). An equilibrium consists of a price schedule q : S × −∞, b →
[0, 1]; government policy functions B : S → −∞, b and D : S × ∞, b → [0, 1]; and a lender
policy function L : S × −∞, b → R; such that: (i) B and D solve the government’s problem
from Section 3.2, conditional on q and L ; (ii) L solves the representative lender’s problem
from Section 3.3 conditional on q and the government’s policy functions; and (iii) market
clearing: equation (7) holds for all s ∈ S and b0 ∈ −∞, b .
Note that the market clearing condition requires that the price schedule clears the market
at all potential b0 , even for debt position that occur off equilibrium. This is a perfection
requirement which ensures that if the government were to deviate from B and issue a suboptimal amount of debt, these bonds would be priced in a manner consistent with equilibrium
behavior going forward.
3.5
Equilibrium Selection
A primary target of the analysis is to explore quantitatively the role of self-fulfilling debt
crises. We do so by exploiting the multiplicity of equilibria. The multiplicity we focus on
is static and involves the current price schedule offered to the sovereign. With an adverse
price schedule, the sovereign can be induced to default today while with a generous price
schedule it would not default for the same debt and output state variables. Moreover, the
22
current and future default behavior of the sovereign in these two circumstances is consistent
with the lenders optimally choosing to lend the prescribed amount at the prescribed price.
We refer to a default with such an adverse price schedule as a rollover crisis. We refer to the
case when the lender chooses to default with the generous schedule as a solvency default.
Note that both rollover crisis and solvency defaults will feature failed auctions, defined
as when the government is unable to auction any positive amount of debt. This is because
no lender will lend at a positive price q if default is certain. Formally, this means that
q(s, b0 ) = 0 for all b0 > (1 − λ)b. In this event, the government must meet its debt obligations
exclusively out of current endowment without recourse to new debt issuances.
To see how this can occur in equilibrium, consider the government’s problem at settlement
following a failed auction. In the case that b0 = (1 − λ)b, which implies that the government
did not issue new bonds when faced with a price of zero, then:
V R (s, (1 − λ)b) = u (Y [1 − (r∗ + λ)b]) + βE [V (s0 )|s, b0 = (1 − λ)b] .
If V R (s, (1 − λ)b) ≤ V D (s) then the government will find it weakly optimal to default at
settlement. Moreover, in the equilibrium we consider, V R (s, b0 ) is decreasing in its second
argument, and so default is weakly optimal for any b0 > (1 − λ)b when q(s, b0 ) = 0. On the
other hand, for a given stock of debt and endowment, there may be a positive price schedule
that can also be supported in equilibrium. That is, if q(s, b0 ) > 0 for some b0 > (1 − λ)b then
the government may decide to issue new bonds to help pay off maturing debt and find it
optimal to repay at settlement.
We incorporate these possibilities in a tractable manner by introducing an additional
(sunspot) random variable ρ which coordinates equilibrium behavior. We let ρ take three
values: ρ ∈ {rC , rT , rV }. State rC is the crisis regime, which implies that there is a rollover
crisis if one can be supported in equilibrium. That is, if the aggregate endowment is low
enough and stock of inherited debt high enough such that V R (s, (1 − λ)b) < V D (s), then
a crisis occurs with q(s, b0 ) = 0 for all b0 > (1 − λ)b. If s is such that a crisis cannot be
supported in equilibrium, there is no crisis when rC is realized. In states rT and rV , there is
no rollover crisis this period, but the states differ in the probability of a crisis next period.
Regime ρ = rT is the tranquil regime, in which the probability that next period ρ = rC is
very low. That is, Pr(ρ0 = rC |ρ = rT ) is close to or equal to zero. On the other hand, ρ = rV
is the vulnerable regime, characterized by the fact that a crisis is much more likely next
period. That is, Pr(ρ0 = rC |ρ = rV ) is sufficiently larger than 0 that a crisis is a real threat.
23
We can think of rT and rV as two regimes, one in which the government is relatively safe
from self-fulfilling crises, and one in which a crisis is relatively likely. More precisely, ρ = rT
implies a very probability of a crisis next period, and then, assuming positive autocorrelation
in the regimes, relative safety going forward. Regime ρ = rV implies danger next period, and
given positive autocorrelation, continued danger into the future. The third regime, rC , is
when a crisis occurs as long as debt is high enough relative to output. The random variable
ρ follows a first-order Markov chain.
One restriction we impose on equilibria is that the price schedule q is homogenous of
degree zero in Y . This implies that prices are functions of the ratios of debt and lenders’
wealth to trend endowment, but not of the level of endowment itself (other than through g
and z as conditioning variables). One could conceivably construct equilibria where this is not
the case by allowing creditor beliefs to vary with the level of trend endowment, conditional
on these ratios. We rule these out.
Lastly, with failed auctions and long-term debt the government may have an incentive
to buy back its debt if the price is low enough. This incentive would be quite strong if q = 0
when it does so. In this case purchasing the outstanding long-term debt costs nothing and
the default penalty is avoided. However, if it buys back virtually all of its debt then its
incentive to default will be gone since the continuation payoff under repayment will have
risen; i.e. V R (s, 0) > V R (s, (1 − λ)b). Hence, a lender should be willing to pay the risk-free
price for this last piece of debt and thereby out bid the government for it. To deal with
this conundrum we follow Aguiar and Amador (2014b) and assume that in the case of a
failed auction, when b0 ≤ (1 − λ)b the price of the debt, q(s, b0 ), is high enough to make
the government just indifferent defaulting and not when it both pays off the maturing debt
and buys back some or all of the remaining outstanding debt. With this indifference, we
then assume that the government mixes between defaulting, and not defaulting. The mixing
probability is set to rationalize the assumed price in the case of a buy-back in a failed auction.
With this indifference we can also assume that the government never buys back its debt in
the case of a failed auction. Hence, this discussion concerns out-of-equilibrium outcomes.
4
Calibration
The quantitative model’s parameters are presented in Tables 5 and 6. We now describe how
we calibrate these parameters using Mexico as our reference economy.
24
Technology
For the endowment process, we assume the growth rate process is governed by
gt+1 = (1 − ρg )ḡ + ρg gt + εt+1
where ε ∼ N (0, σg2 ). The transitory component of output zt is assumed to be iid, orthogonal
to εt and to have mean zero and variance σz2 . The implied growth rate of log output is
yt+1 − yt = gt+1 + zt+1 − zt + εt+1 .
We estimate this model using quarterly Mexican GDP expressed in constant US dollars for
the period 1980Q1 through 2015Q1.9 The estimated parameter vector is reported in Table
5.
With these parameters in hand, we discretize the process for g using Tauchen’s method
p
using 50 grid points spanning ±3σg / 1 − ρ2g . The iid z shock is drawn from a continuous
Normal distribution truncated at ±3σz . When taking expectations, we numerically integrate
over z’s continuous distribution by evaluating at 11 grid points.
Preferences
The coefficient of relative risk aversion for the sovereign and the creditors is set to 2.
The government’s discount factor is set through the moment matching procedure described
below.
Financial Markets
We set the risk-free interest rate at 1 percent quarterly (hence, 4 percent annually). The
average maturity length is set to 8 quarters, that is λ = 1/8, which implies a Macaulay
duration of 6.4 quarters. This is shorter than the average maturity (or duration) observed
in many emerging markets. However, maturity length is not constant over time and tends
to shorten when the probability of a crisis is high (Broner, Lorenzoni, and Schmukler, 2013;
Arellano and Ramanarayanan, 2012). Moreover, much of a country’s short-term debt is
9
We use the dollar-value of GDP to capture the fact that a depreciation of the peso makes servicing dollardenominated debt more costly in terms of domestic goods, and hence this additional source of variation is
relevant for the sovereign’s borrowing and default decisions. Using constant US dollars makes the US
consumption basket the numeraire, which is a reasonable basis for our real model. As sovereign bonds are
not typically indexed to inflation, our model does not incorporate the fact that surprise dollar inflation
represents a windfall to debtors. However, given the low and stable US inflation over this period, this is not
likely to be quantitatively important.
25
issued domestically (whether in dollars or local currency), and thus is not reflected in the
average maturity of external debt. For simplicity, our model only has external debt of constant maturity, raising the question of how to accurately capture a world in which maturity
varies over time and the amount due (and to whom) in any given quarter is not uniform.
Given our focus on crises, we set the average maturity length to a value that is relatively
short.
The endowment while in default is Y D (s) = (1−φ(s))Y (s), where Y (s) is the endowment
absent default. In particular, we set::
Y D (s) =

Y (s)
if default occurs this period
(1 − d)ez̄−z(s) Y (s)
otherwise
where d ∈ (0, 1) is the adjustment to income due to being in default status. A few things are
of note in this specification. One is that the cost of default is linear in output, as in Aguiar and
Gopinath (2006). In contrast, Arellano (2008) introduced a non-linear cost of default which
made default disproportionately more costly in good endowment states and “forgiven” – at
least in terms of output costs –in low endowment states. The Arellano specification amplifies
the impact of endowment fluctuations in the decision to default while also making default a
better insurance option in low-endowment states. This helps the model generate additional
volatility of spreads and frequency of default, but does so by making endowment risk more
important rather than less. The empirical facts outlined above and in complementary work
like Tomz and Wright (2007) suggests that this pulls the model in the wrong direction relative
to the data. The parameter d is set below by matching moments.
A second feature of φ(s) is that no output is lost in the initial period of default. This
is not a crucial assumption given the ability to adjust d, but does simplify the exposition of
how growth is correlated with the switch to default status. The timing is as if default occurs
at the end of a quarter and output is not affected until the next quarter.
Finally, for computational convenience, we fix the transitory income shock to be a constant z̄ while in default. Thus Y (s) is adjusted by ez̄−z(s) , where z(s) is the transitory shock
associated with state s. Given that d is a free parameter, the level of z̄ can be normalized
to 0 without loss.
In addition to lost output, default also brings exclusion from financial markets. We set
the re-entry probability after default to 0.125 quarterly; that is, the average duration of
26
default is two years. This is in the range documented by Gelos, Sahay, and Sandleris (2011)
for the 1990s, but lower than Tomz and Wright (2013)’s median of 6.5 years using a much
longer sample.
We assume that the wealth of the available pool of creditors (relative to the economy’s
endowment) follows an AR(1):
wt+1 = (1 − ρw )w̄ + ρw wt + ut+1 ,
where u ∼ N (0, σw2 ) and iid over time and orthogonal to the other shocks. In the model,
shocks to w generate fluctuations in the risk premium on sovereign bonds conditional on the
other state variables. Based on the data presented in Section 2 and the related work in the
finance literature, we use fluctuations in the price-earnings ratio of the S&P 500 as a proxy
of the empirical appetite for risk. Specifically, the autocorrelation of the P/E ratio between
1993Q4 and 2014Q4 is 0.91. We therefore set ρw = 0.91. The remaining parameters, (w̄, σw )
are set below by matching moments from the model. We discretize the AR(1) process using
p
5 grid points spanning ±3σw / 1 − ρ2w .
Creditor Beliefs
We calibrate creditor beliefs as follows. Recall that beliefs are characterized by a threestate Markov regime-switching process. One regime (“crisis”) is the realization of a rollover
crisis that generates default. The remaining regimes (“tranquil” and “vulnerable”) differ
in the probability of the crisis regime. We estimate a two-state regime-switching model
to Mexico’s EMBI spread series from 1993Q4-2014Q4. The rationale for the two-regime
estimation rather than the model’s three regimes is the absence of default in the sample
period. We estimate a transition probability of moving from the low-spread to the highspread regime of 0.026, and the reverse transition of 0.177. The time series of the EMBI
spread and the associated probability of being in the vulnerable regimes are depicted in
Figure 3. We can see that the Mexico crisis of 1994-95 and the spillover from the Russia
crisis of 1998 are evident in spreads and the probability of being in the “vulnerable” regime.
The 2008 spike is spreads is not interpreted as a shift in regimes.
In estimating the regime-switching model, we would like to separate movements in creditor beliefs from movements in risk premia. For this reason, we have also estimated a
regime-switching model for Mexico which includes the S&P P/E ratio as an explanatory
variable, with a coefficient that varies by regime. The respective transition probabilities for
27
this augmented model are nearly identical (0.026 and 0.176, respectively). If we subtract
a linear time-trend from the spread series, the transition probabilities are 0.061 and 0.077,
which implies slightly more time in the vulnerable regime.
In the calibrated model, we therefore set the transition probabilities at 0.028 and 0.12
[to be updated]. This leaves the probability of a rollover crisis in the vulnerable regime.
We set this at 20 percent, and perform robustness checks regarding this parameter. The
probability of transiting from the tranquil regime to a crisis is set to the product of transiting
to vulnerable and the probability of a crisis conditional on vulnerable; that is, we allow
the regime to transit from tranquil to crisis via vulnerable within the same quarter. This
probability is nearly zero: (0.028)(0.20) = 0.006.
0
0
.2
.4
.6
Regime Probability
.8
Sovereign Spread (Basis Points)
500
1000
1
1500
Figure 3: Sovereign Spread Regimes for Mexico
1995q1
2000q1
2005q1
EMBI Spread
2010q1
2015q1
Prob. Vulnerable
Simulated Matched Moments
We calibrate the remaining moments by simulating the model and matching targeted
empirical moments. Specifically, the remaining parameters are the government’s discount
factor, the proportion of output lost during default, and the mean and variance of creditor
wealth-to-GDP (w̄, σw ).
For Mexico, we have external debt to annual GDP (both in US dollars) for the period
28
2002Q1 through 2014Q3. The average over this period is 16.4 percent, which translates into
a quarterly debt-to-income ratio of 65.6 percent. This measure of debt includes external debt
by the government as well as banks. A longer time series exists for a narrower stock of debt
issued by the federal government. This series suggests that debt levels were higher in the
1990s and have been falling in the 2000s and 2010s. Hence our measure of 65 percent may
be an understatement. The average EMBI spread for Mexico over the entire period is 3.4
percent. These moments are natural targets as their simulated counterparts are particularly
sensitive to the government’s discount factor and the default cost.
The spread is a combination of risk premium and default probability. To help separately
identify these, we also target a frequency of default of 2 percent per annum, which is in line
with the facts documented by Tomz and Wright (2013). An additional moment related to
the role of risk premia in spreads is based on the regression of spreads on P/E ratios reported
in Table 4. In particular, this regression has an R2 of 0.29. The model counterpart is the
R2 of a regression of the spread on creditor wealth (which is our proxy for risk aversion).
In the simulated model, the mean debt-to-income ratio and the spread is conditional on
not being in the default state. More specifically, we compute the mean conditional on being
out of the default state for at least 25 quarters. The reason we condition on being in good
credit standing for a significant period of time is that the government exits default status
with zero debt. Zero debt after default is not a realistic feature of the model and hence we
focus on the ergodic distribution conditional on having sufficient time to rebuild debt.
These four targets, the model counterparts, and the associated parameters are reported
in Table 6.
4.1
Solution
The fact that u and v are homogenous functions and the budget set for the government’s
problem is homogenous of degree one in Y implies that the level of endowment Y is not a
relevant state variable for the equilibrium price schedule and associated policies, where recall
that the policy functions for debt issuance and bond demand have been defined as ratios to
current endowment Y .10 Therefore we may solve for an equilibrium price schedule and the
associated government’s and lender’s problems on a truncated (“detrended”) state space that
10
See Alvarez and Stokey (1998) for a formal treatment of dynamic programming with homogenous functions.
29
Table 5: Parameters I: Set Prior to Simulation
Parameter
Value
Source
Endowments:
(1 − ρg )ḡ
ρg
σg
σz
0.0034
0.445
0.012
0.003




2
2
Standard
Standard
Financial Markets:
Quarterly Risk-free rate (r∗ )
Reciprocal of Avg. Maturity (λ)
Default Re-entry Prob (ξ)
Autocorrelation of Creditor Wealth (ρw )
0.01
0.125
0.125
0.91
Standard
N/A
Gelos et all (2011)
P/E Ratio AR(1) Coeff
Belief Regimes:
Pr(ρ0 = rV |ρ = rT )
Pr(ρ0 = rT |ρ = rV )
Pr(ρ0 = rC |ρ = rV )
Pr(ρ0 = rC |ρ = rT )
0.028
0.120
0.200
0.006
Preferences:
Sovereign CRRA (σ)
Creditor CRRA (γ)
Mexico GDP Data
1980Q1-2015Q1



Mexico EMBI Data
1994Q1-2014Q4
N/A
N/A
omits Y . With these policies in hand, we can simulate the economy by drawing a sequence
{gt , zt }∞
t=0 and then iterating on the detrended policy functions to obtain equilibrium paths
for debt-to-income as well as prices and default outcomes. Finally, the path of endowment Yt
can be constructed from {gt , zt }∞
t=0 , and the level of any endogenous variable can be obtained
by scaling up its ratio with income.
30
Table 6: Parameters II: Simulated Method of Moments
5
5.1
Target Moment
Data
Model
Debt-to-Income (Quarterly)
Mean Spread (Annual)
Default Frequency (Annually)
R2 Reg of Spread on Risk Measure
65.6%
3.4%
0.02
0.29
66.1%
3.3%
0.02
0.25
Parameter
Value
Discount factor (β)
Default Cost (d)
Mean Creditor Wealth Relative to Y (w̄)
Std Dev Creditor Wealth (σw )
0.835
0.068
2.53
2.64
Quantitative Results
Equilibrium Prices
In this section we characterize the equilibrium bond price schedules and policy functions for
debt issuance. In Figure 4a we plot the price schedule q as a function of b0 given that (g, z, w)
are set at their mean values and beliefs are “Tranquil” (ρ = ρT ). Recall that we normalize
the coupon payment so that the price of a risk free bond is one.
The dominating feature of the price schedule is the extreme nonlinearity once default
becomes a likely scenario. That is, in this region, the probability of default next period is
sensitive to the choice of how much debt to issue. This is a well known feature of these
models, as discussed in Aguiar and Gopinath (2006) and Aguiar and Amador (2014a). The
decision to issue debt in this region has an average cost q but also a marginal cost related
to ∂q/∂b0 , which quickly becomes large in the nonlinear range.
Recall that we target the conditional mean of b at 0.69. This is close to the start of the
extremely nonlinear portion of the price schedule. To get a better sense of the relevant state
space, Figure 5 plots the histogram of b in the simulated model, conditional on being in good
credit standing for at least 25 quarters.The middle 98th percentiles spanning [0.665, 0.709].
31
This reflects the fact that impatience drives the sovereign to accumulate debt, but the nonlinearity of the price schedule places a limit on leverage. The ergodic distribution therefore
clusters around this region.
In Figure 4b plots three price schedules over this domain conditional on w being at its
mean and ρ = ρT , and with each schedule depicting an alternative g realization. In particular,
we depict schedules associated with the mean g as well as plus and minus 3σg /(1 − ρ2g ). Note
that as g increases, the price schedule shifts up and flattens. This generates a pattern in which
high growth leads to more borrowing, as the average and marginal cost of debt is relatively
low. This feature of the model is reminiscent of Aguiar and Gopinath (2006), which presents
a quantitative Eaton-Gersovitz model in which the source of risk is a stochastic growth
process.
Figure 4c explores the response of equilibrium prices given different creditor wealth levels
(conditional on g at its mean and ρ = ρT ). Note that for high levels of b0 , a high level of
creditor wealth raises q as creditors are more willing to tolerate risk. Interestingly, for lower
levels of debt, a positive wealth shock lowers prices. This is due to a disciplining effect.
A low level of current wealth, given persistence, deters the sovereign from accumulating a
lot of wealth in the future. Therefore, the incentive to dilute today’s new bondholders is
mitigated, raising today’s price. This effect depends on the fact that bonds have a maturity
greater than one period and that the wealth state is persistent.
Figure 4d depicts the shift in price schedule between the tranquil and vulnerable regimes.
The shift in creditor beliefs to Vulnerable generates a sharp drop in prices, reflecting the
higher probability of a rollover crisis in the near term.
32
Figure 4: Equilibrium Price Schedules
(a) Equilibrium Price Schedule (Tranquil,Mean g, w)
(b) Alternative g
q(b0 ; :) 0.960
0
q(b ; :) 1.000
0.955
0.900
0.950
0.945
0.800
0.940
0.700
0.935
0.600
0.930
0.500
0.925
0.400
0.920
0.635
0.640
0.645
0.650
0.655
0.300
0.200
0.665
0.670
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.675
0.680
0.685
b0
<g
1 ! ;2g
g = g7
<g
g = g7 + 3 q
1 ! ;2g
0.100
0.000
0.000
0.660
g = g7 ! 3 q
0.900
b0
(c) Alternative w
(d) Alternative Beliefs
q(b0 ; :) 0.960
q(b0 ; :) 0.960
0.955
0.955
0.950
0.950
0.945
0.945
0.940
0.940
0.935
0.930
0.935
0.925
0.930
0.920
0.635
0.640
0.645
0.650
0.655
0.660
0.665
0.670
<w
w=w
7 ! 3p
1 ! ;2w
w=w
7
<w
w=w
7 + 3p
1 ! ;2w
0.675
0.680
0.925
0.685
b0
0.920
0.635
0.640
0.645
0.650
0.655
0.660
0.665
; = ;T
; = ;V
33
0.670
0.675
0.680
0.685
b0
40
30
Density
20
10
0
.62
.64
.66
Debt/Y
.68
.7
kernel = epanechnikov, bandwidth = 0.0005
Figure 5: Ergodic Distribution of Debt-to-Income
5.2
Policy Functions
In Figure 6 we plot the policy function for debt issuances as a function of inherited debt and
for two belief regimes. We integrate over the shocks to g and w using the ergodic distribution.
We plot the policy functions over the relevant range of debt from the ergodic distribution
conditional on being out of the default state for at least 25 quarters.
Two striking facts emerge from the figure. First, debt policy functions are quite flat
around the 45 degree line: The optimal policy features sharp leveraging and deleveraging
that offsets the impact of good and bad growth shocks, respectively, and returns b0 to the
neighborhood of the crossing point quite rapidly. The second fact is that the switch to the
Vulnerable regime leads to rapid deleveraging (or default). The jump in spreads provides
an incentive to de-lever, a feature reminiscent of Cole and Kehoe (2000). The ergodic
distribution by belief regime is depicted in Figure 7. The shift down in the debt-issuance
policy function in the Vulnerable regime is reflected in the leftward shift of the conditional
distribution of debt.
6
Equilibrium Outcomes
[Still to be updated]
34
.6
.62
.64
b'
.66
.68
.7
Figure 6: Policy Functions
.6
.62
.64
.66
.68
.7
b
Vulnerable
Frequency
Tranquil
45 Degree
.62
.64
.66
Debt/Y
Tranquil
.68
.7
Vulnerable
Figure 7: Ergodic Distribution of Debt-to-Income by Belief Regime
In this section we lay out the results of our benchmark model in order to both understand
its mechanics and how it compares to the data we discussed. We will also be interested in
comparing the results of our benchmark model to various other permutations which we
discuss later. For now, focus on the first column where we report the basic statistics from
our benchmark model. The first three statistics, which were targeted, match the long-run
35
data for Mexico and is in the ball park for other emerging economies. The correlation of
the average excess return and the growth rate of output also seems in the neighborhood of
the other emerging markets in that the correlation is quite weak, as it is in the data. The
notable deviation relative to facts is that the volatility of spreads, which is very low in the
model. We report the coefficient of variation and this should be roughly 1.00 to match the
data. We will come back to this point later.
Table 7: Benchmark and Alternative Models of Beliefs
B/Y
Default Freq.
E {r − r∗}
σ (r − r∗)
corr(r − r∗, ∆y)
Benchmark
Model
0.69
0.01
0.02
1.5×10−3
-0.16
Tranquil
Only
TBA
TBA
TBA
TBA
TBA
Vulnerable
Always
TBA
TBA
TBA
TBA
TBA
The ergodic distribution of debt-to-income by belief regime is depicted in Figure 6. The
tranquil regime is both more frequent and has a higher average debt level relative to the
vulnerable regime. The crisis regime is the least frequent belief state. This figure reflects the
impact of the current growth shock on the debt ratio. The value of Bt /Yt−1 is distributed
much more tightly, as indicated by the debt policy functions. Note that higher debt levels
do occur in the vulnerable and crisis regimes, just as lower debt levels arise in the tranquil
regime. These occur during transitions between the tranquil and either of the two risky
regimes (vulnerable or crisis).
The associated distribution of interest rate spreads is depicted in Figure 8. Higher spreads
are almost always associated with our risky regimes - vulnerable or crisis - and low spreads
are almost always associated with the tranquil regime. However a careful eye can detect
some lower spread values for the risky regimes. These arise from high growth rate shocks
that lower the debt-to-GDP ratio and hence spreads. Figure 9 displays the distribution of
the change in spreads. We can see that large upward jumps in spreads tend to happen in the
vulnerable regime while large drops in the tranquil regime. Sometimes, large upward jumps
in the spread can happen in the tranquil regime if the growth realization turns out to be
particularly bad.
This spread can be decomposed into a default premium and a risk premium. Specifically,
36
Frequency
Figure 8: Ergodic Distribution of Spreads
.025
.03
.035
Spread
Tranquil
.04
.045
Vulnerable
Frequency
Figure 9: Ergodic Distribution of Change in Spreads
-.02
-.01
0
Quarterly Change in Spread
Tranquil
.01
.02
Vulnerable
the risk premium is the standard difference between the expected implied yield on sovereign
bonds and the risk free interest rate. The default premium is the promised yield that would
equate the expected return on sovereign bonds (inclusive of default) to a risk-free bond; that
is, the yield that would leave a risk-neutral lender indifferent. Panel (a) of Figure 10 depicts
37
the risk premium and Panel (b) depicts the default premium.
Figure 10: Decomposition of Spread
(a) Risk Premium
Frequency
Frequency
(b) Risk-Neutral Spread
0
.005
.01
.015
Risk Premia
Tranquil
.02
.025
.01
.02
Vulnerable
.03
Risk Neutral Spread
Tranquil
.04
.05
Vulnerable
Higher defaults spreads and risk premia are associated with our vulnerable regimes, while
lower values occur under the tranquil regime. The correlation of the default spread and the
risk premia is high, indicating that default risk is driving up both components of the spread.
Figure 11: Interest Rate Crises
0
10
Density
20
Frequency
30
40
(a) Distribution of Growth Rates: Crises and(b) Frequency of Crisis Growth Rates: By Belief Regime
Non-Crises
-.05
0
Quarterly Growth
Change in Spread<=95th
.05
-.04
Change in Spread>95th
-.02
0
Quarterly Growth
Tranquil
.02
.04
Vulnerable
Figure 11 Panel (a) reports the distribution of growth rates during crisis and non-crisis
periods. In the model, there is a clear tendency for crisis to happen during low growth
periods but it is also the case the crises occur when the growth rate is above its mean. These
crises during “good times” are crises that accompany a shift from Tranquil to Vulnerable
regimes.
38
Table 8: Interest Rate Crises
Tranquil
Vulnerable
Share by
Regime
51.60
48.40
Share with
g Collapse
4.19
2.25
Share with
z Collapse
1.03
0.65
Share with Belief
Change to ρV
27.94
Table 9: Defaults
Tranquil
Vulnerable
Crisis
Share by
Regime
Share with
g Collapse
Share with
z Collapse
47.16
9.79
43.05
45.03
7.60
12.03
4.13
0.44
2.25
What if
Tranquil
(Counterfactual)
47.16
3.19
0.00
What if
Always Tranquil
(Counterfactual)
1.15
0.00
0.00
Share with Belief
Change from
ρt−1 = ρC
8.54
21.38
Figure 11 Panel (b) depicts the frequency of crisis at different growth rates for each
type of regime. For this purposes of this graph, a crisis is defined as jump up in spreads
that are in 95th percentile of the distribution shown in Figure 9. Table 8 provides some
crisis statistics. Crisis are somewhat more likely in a Vulnerable regime as compared to a
Tranquil regime (the jump in spread in the Crisis regime is not reported because it is infinite
as the sovereign defaults in the Crisis regime). To see how important growth shocks were
to crisis, we report the fraction of crises in each regime caused by a 3 standard deviation
drop in g from the previous period or a a negative 1 standard deviation shock to z, episode
we refer to as “collapses.” Growth collapses account for close to one-half of all crises in
the Tranquil regime, although crises happen for less extreme shocks.11 In contrast, growth
collapses account for a less than a fifth of the crises in the Vulnerable regime. Instead, crises
tend to be triggered by the change in beliefs that accompany the shift from tranquillity to
vulnerability.
Figure 12 and Table 9 reports some default related properties and statistics for our
11
A good chunk of crisis happen when the transitory shock z is low. Since transitory shocks do not cause
adverse shifts in the pricing function for debt, the sovereign has the motivation as well as the opportunity to
smooth through the shock by borrowing. Such “consumption-smoothing” induced borrowing, if large, can
cause the interest rate on debt to jump up.
39
Frequency
Figure 12: Distribution of Defaults by Growth and Regime
-.06
-.04
-.02
Quarterly Growth
Tranquil
Rollover Crisis
0
.02
Vulnerable
benchmark economy. From the table one can see that each regime accounts for roughly one
third of the overall defaults. One can see that virtually all of the defaults in the tranquil
regime were accompanied by a growth collapse, while only 60-70 percent of defaults were in
the non-tranquil regimes.
Table 9 also presents some statistics to gauge the importance of beliefs to our defaults.
We report how many of them would have taken place if the regime was tranquil. All of the
tranquil defaults would still occur by construction, however virtually none of the vulnerable
or risky defaults would have occurred. To indicate that in some sense even the tranquil
defaults are a product of beliefs we look at how many of them would have occurred if beliefs
were always tranquil. The answer is none of them.
This result can be understood in the context of Figure 13 which plots default indifference
curves for our three regimes and also for the case in which beliefs were always tranquil or
always vulnerable. Because both the continuation payoffs of the sovereign and the price
schedules offered by the lenders are effected by the different belief configurations, the default
indifference curves as a function of the debt-to-output level at the end of the last period and
the growth shocks end up being close to parallel shifts of each other. This results the fact
that there is essentially a fixed realized debt-to-output regime which triggers default due to
the low correlation in growth shocks.
Finally, to gauge how important shifts in beliefs were to our defaults, we report the
share of defaults that were accompanied by a belief shift. Because a shift to the tranquil
40
regime shifts out the default indifference curve, essentially none of our defaults in the tranquil
regime occurred with a shift in beliefs. Instead tranquil defaults occur because the country
has increased its debt level in response to being in the tranquil regime. In stark contrast,
the bulk of the defaults in the vulnerable regime occurred immediately after a shift into the
vulnerable regime from the tranquil regime. Since a shift from tranquil to crises beliefs is a
very low probability event, very few of our defaults occurred in the crises regime with a shift
in beliefs from tranquil.
Figure 13: Default Indifference Curves by Belief Regime
Indifference Curves
0.85
Bt/Yt-1
0.8
0.75
0.7
Solvency (AT)
Solvency (Tranquil)
Solvency (Vulnerable)
Liquidity (Vulnerable)
0.65
0.6
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
gt
In closing we stress two important findings. First, as shown in Figure 10 our model
generates the sort of patterns we saw in the data regarding crisis and non-crisis periods.
Crises periods tend to be periods of lower growth but crisis regularly also happen when
fundamentals (as measured by the growth in output) seem good. Thus, we largely reproduce
this key pattern in the data.
Second, as show in Table 7, our Markov structure on beliefs generates time variation in
41
default risk and that helps to get time variation in spreads. If we compare our benchmark
results to a model, along the lines focused on by much of the literature which features
the most generous possible price schedule (called the always tranquil) we see a much lower
default frequency, average spread, and volatility of the spread. Moreover, the correlation of
the excess return and the growth rate increases. These results arise because output shocks are
relatively controllable risk, and the tendency of the price of default risk to increase sharply
with the level of the default risk strongly encourages countries to adjust their debt-to-output
ratio so as to keep these risks at constant low level. If we compare our benchmark model
to a quantitative version of the original Cole-Kehoe model (called vulnerable always) which
features constant risk of a crisis, we see that the debt-to-output ratio falls in response to this
high level of risk, while the frequency of default and the average spread increases. However,
the constancy of the risk of a rollover crisis leads to a fall in the relative volatility of the
spread. Thus, the time variation in the nonfundamental risk of default coming from our
Markov structure is helping the model match the data. But clearly more needs to be done
in this dimension as the model is off by an order of magnitude.
7
Crises
[To be Updated]
In this section, we explore sovereign debt crises viewed through the context of the model.
In the spirit of our empirical exercise, we first consider the economy’s behavior conditional
on a sharp increase in spreads. We do this in the Tranquil Regime as well as the Vulnerable
Regime. These event studies are unconditional on what underlying shock triggers the increase
in spreads. To shed light on the role of specific shocks, we then consider the responses to a
large decline in income growth as well as a shift in creditor beliefs.
7.1
Crises in the Tranquil Regime
In this section, we consider debt crises conditional on Tranquil Regime beliefs. Specifically,
we condition on an increase in spreads greater than xx during periods when ρ = rT . The
threshold for a crisis is the 95th percentile of the unconditional ergodic distribution of the
change in spreads conditional on repayment. We stress that the crisis analysis conditions
on repayment; by construction, quarters in which the sovereign defaults are associated with
42
an infinite spread. Recall that the empirical analysis included two default episodes, so the
model analysis represents a slight departure in this regard.
Figure 14 depicts the path of spreads (Panel a) and the implied short-long yield curve
(Panel b) conditional on a significant increase in spreads between periods t = −1 and t = 0
and ρ0 = rT . Note that the period before the crisis is one of relatively low spreads. This
reflects an accumulation in debt as a fraction of output leading up to the crisis, as a high
debt-to-income ratio is necessary to sustain the subsequent high spread. Panel (b) indicates
the increase in one-period spreads is greater than the long (benchmark maturity) spread.
This reflects that short-run default risk exceeds the longer run risk due to mean reversion.
Figure 15 depicts the average growth rate of the economy conditional on a crisis at time
t = 0. Note that growth experiences a slight boom before the crisis, and then a sharp bust.
This reflects the boom-bust nature of crises absent shifts in creditor beliefs. The crisis itself
is triggered by a large decline in economic growth. Given the ex ante boom, this large decline
is relatively unexpected. This reflects that the sovereign would like to avoid states in which a
crisis is likely, and thus periods with a large spike in spreads in the Tranquil Regime requires
an unexpected deterioration in fundamentals.
[to be added: path of debt; default probability versus risk premia]
43
Figure 14: Tranquil-Regime Crises: Mean Spreads
(a) Mean Spread
T Crisis at t = 0
0.12
0.115
E[ Spread | No Default]
0.11
0.105
0.1
0.095
0.09
0.085
0.08
0.075
-10
-5
0
5
10
15
20
25
30
20
25
30
Time (Quarters)
(b) Short-Long Spread
T Crisis at t = 0
1.8
1.7
E[ Yield Curve | No Default]
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
-10
-5
0
5
10
15
Time (Quarters)
44
Figure 15: Tranquil-Regime Crises: Mean Growth
T Crisis at t = 0
0.03
E[ gt | No Default]
0.02
0.01
0
-0.01
-0.02
-0.03
-10
-5
0
5
10
15
20
25
30
Time (Quarters)
7.2
Vulnerable-Regime Crises
We now consider a similar event study but conditional on Vulnerable-Regime beliefs. Figure
16 depicts the path of spreads and the implied yield curve conditional on an increase in
spreads at time t = 0 and ρ0 = rV . Comparing Figure 16 to 14 we see that conditional on
meeting the crisis threshold, the average increase in spreads is much greater in the Vulnerable
Regime.
Moreover, the ex ante decline in spreads is much smaller in the Vulnerable-Regime crisis.
This reflects a number of differences between crises in the two alternative belief regimes.
Under Vulnerable beliefs, default is likely at lower debt levels due to the high probability of
a rollover crisis. Thus high spreads do not need a large run up in debt beforehand. Moreover,
a change in beliefs from Tranquil to Vulnerable will by itself induce an increase in spreads.
Our calibration sets the probability of this transition at 2 percent. Therefore, belief shifts to
45
Vulnerable are always surprises. Table 8 indicates that more than half of the crisis episodes
conditional on Vulnerable Regime are associated with a shift from Tranquil to Vulnerable
beliefs.
Figure 17 plots the path of growth around the crisis event. Similar to spreads, we see
a distinction in the average growth relative to the Tranquil Regime’s behavior depicted in
Figure 15. Growth does not boom as much leading up to the Vulnerable-Regime crises and
does not decline as much during the crisis. This is consistent with the above discussion on
belief changes; the surprise in a Vulnerable-Regime crisis is due to shifts in creditor beliefs,
not unexpected declines in income. It is also sheds light on Figure 11. That figure indicated
that Vulnerable-Regime crises were associated with somewhat higher growth rates than the
Tranquil-Regime crises; Figure 17 indicates that the Vulnerable crises are also associated with
lower growth ex ante, and thus a smaller decline in growth during the periods surrounding
a crisis.
[to be added: path of debt; default probability versus risk premia]
46
Figure 16: Vulnerable-Regime Crises: Mean Spreads
(a) Mean Spread
V Crisis at t = 0
0.12
0.115
E[ Spread | No Default]
0.11
0.105
0.1
0.095
0.09
0.085
0.08
0.075
-10
-5
0
5
10
15
20
25
30
20
25
30
Time (Quarters)
(b) Short-Long Spread
V Crisis at t = 0
1.8
1.7
E[ Yield Curve | No Default]
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
-10
-5
0
5
10
15
Time (Quarters)
47
Figure 17: Vulnerable-Regime Crises: Mean Growth
V Crisis at t = 0
0.03
E[ gt | No Default]
0.02
0.01
0
-0.01
-0.02
-0.03
-10
-5
0
5
10
15
20
25
30
Time (Quarters)
7.3
Growth Crises
Our discussion of crises that occur under Tranquil-Regime beliefs highlighted the importance
of boom-bust dynamics for output. We now explore the economy’s behavior conditional on
a large negative realization to output growth. Specifically, our event study is conditional
on a growth realization three standard deviations below average in period t = 0. Figure 18
plots the path of income growth conditional on the realized innovation to the growth process
exceeding xx at t = 0. The low persistence of growth is reflected in the quick recovery of
growth to its unconditional mean.
Figure 19 depicts the response of spreads conditional on Tranquil-Regime beliefs (Panel a)
and Vulnerable-Regime beliefs (Panel b). Panels (c) and (d) depict the respective response to
the Long-Short spread. The figure indicates that the impact of a negative growth innovation
on spreads is amplified by Vulnerable-Regime beliefs. This indicates that the sensitivity of
48
Figure 18: Growth Innovation
Large Negative g at t = 0
0.03
E[ gt | No Default]
0.02
0.01
0
-0.01
-0.02
-0.03
-10
-5
0
5
10
15
20
25
30
Time (Quarters)
default to fundamentals is enhanced by the high probability of a rollover crisis.
[to be added: path of debt; default probability versus risk premia]
49
Figure 19: Growth Crises: Spreads
(a) Spreads in Tranquil Regime
(b) Spreads in Vulnerable Regime
Large Negative g and T-Regime at t = 0
Large Negative g and V-Regime at t = 0
0.12
0.12
0.115
0.115
0.11
E[ Spread | No Default]
E[ Spread | No Default]
0.11
0.105
0.1
0.095
0.09
0.085
0.105
0.1
0.095
0.09
0.085
0.08
0.08
0.075
0.075
-10
-5
0
5
10
15
20
25
30
-10
-5
0
Time (Quarters)
1.7
1.7
1.6
1.6
1.5
1.4
1.3
1.2
1.1
1.2
1.1
1
20
25
0.8
-10
30
Time (Quarters)
7.4
1.3
0.9
15
30
1.4
0.9
10
25
1.5
1
5
20
Large Negative g and V-Regime at t = 0
1.8
E[ Yield Curve | No Default]
E[ Yield Curve | No Default]
Large Negative g and T-Regime at t = 0
0
15
(d) Short-Long in Vulnerable Regime
1.8
-5
10
Time (Quarters)
(c) Short-Long in Tranquil Regime
0.8
-10
5
-5
0
5
10
15
20
25
30
Time (Quarters)
Belief Crises
We now explore a shift in creditor beliefs. Specifically, we condition on a shift in beliefs from
Tranquil at t = −1 to Vulnerable at t = 0. Figure 20 plots the response of spreads and
the yield curve. As anticipated by the discussion of crises conditional on Vulnerable-Regime
beliefs, a negative shift in beliefs generates a large increase in spreads.
[to be added: path of debt; default probability versus risk premia]
50
Figure 20: Belief Crises: Change from Tranquil to Vulnerable
(a) Spreads
(b) Short-Long Yield Curve
T to V at t = 0
1.8
1.7
0.11
1.6
E[ Yield Curve | No Default]
E[ Spread | No Default]
T to V at t = 0
0.12
0.115
0.105
0.1
0.095
0.09
0.085
1.4
1.3
1.2
1.1
0.08
1
0.075
0.9
-10
-5
0
5
10
15
20
25
0.8
-10
30
Time (Quarters)
8
1.5
-5
0
5
10
15
20
25
30
Time (Quarters)
Conclusion and Directions for Future Research
[to be added]
51
References
Aguiar, M., and M. Amador (2014a): Handbook of International Economics Volume
4 chap. Sovereign Debt, pp. 647–687. North-Holland Elsevier.
(2014b): “Take the Short Route: How to Repay and Restructure Sovereign Debt
with Multiple Maturities,” working paper.
Aguiar, M., and G. Gopinath (2006): “Defaultable Debt, Interest Rate and the Current
Account,” Journal of International Economics, 69, 64–83.
(2007): “Emerging Market Business Cycles: The Cycle is the Trend,” Journal of
Political Economy, 115, 69–102.
Alvarez, F., and N. Stokey (1998): “Dynamic Programming with Homogenous Functions,” Journal of Economic Theory, 82, 167–189.
Arellano, C. (2008): “Default Risk and Income Fluctuations in Emerging Markets,”
American Economic Review, 98(3), 690–712.
Arellano, C., and A. Ramanarayanan (2012): “Default and the Maturity Structure
in Sovereign Bonds,” Journal of Political Economy, 120, 187–232.
Auclert, A., and M. Rognlie (2014): “Unique Equilibrium in the Eaton-Gersovitz
Model of Sovereign Debt,” working paper.
Borri, N., and A. Verdelhan (2011): “Sovereign Risk Premia,” MIT Sloan Working
Paper.
Broner, F., G. Lorenzoni, and S. Schmukler (2013): “Why do emerging economies
borrow short term?,” Journal of the European Economics Association, 11, 67–100.
Calvo, G. (1988): “Servicing the Public Debt: The Role of Expectations,” American
Economic Review, 78, 647–661.
Chatterjee, S., and B. Eyigungor (2012): “Maturity, Indebtedness, and Default Risk,”
American Economic Review, 102, 2674–99.
Cole, H., and T. Kehoe (2000): “Self-Fulfilling Debt Crises,” The Review of Economic
Studies, 67, 91–116.
52
Eaton, J., and M. Gersovitz (1981): “Debt with Potential Repudiation: Theoretical
and Empirical Analysis,” The Review of Economic Studies, 48(2), 289–309.
Gelos, G., R. Sahay, and G. Sandleris (2011): “Sovereign Borrowing by Developing
Countries: What Determines Market Access?,” Journal of International Economics, 83,
243–254.
Hatchondo, J. C., and L. Martinez (2009): “Long-Duration Bonds and Sovereign
Defaults,” Journal of International Economics, 79, 117–125.
Leland, H. (1994): “Bond Prices, Yield Spreads, and Optimal Capital Structure with
Default Risk,” IBER Finance Working Paper 240.
Longstaff, F., J. Pan, L. Pedersen, and K. Singleton (2011): “How Sovereign is
Sovereign Credit Risk,” American Economic Journal: Macroeconomics, 3, 75–103.
Lorenzoni, G., and I. Werning (2013): “Slow Moving Debt Crises,” NBER Working
Paper 19228.
Neumeyer, P., and F. Perri (2005): “Business Cycles in Emerging Economies: The Role
of Interest Rates,” Journal of Monetary Economics, 52, 345–380.
Pan, J., and K. Singleton (2008): “Default and Recovery Implicit in the Term Structure
of Sovereign CDS Spreads,” Journal of Finance, 63, 2345–2384.
Reinhart, C., K. Rogoff, and M. Savastano (2003): “Debt Intolerance,” Brookings
Papers on Economic Activity, 2003, 1–74.
Tomz, M., and M. Wright (2007): “Do Countries Default in Bad Times?,” Journal of
the European Economics Association, 5, 352–360.
(2013): “Empirical Research on Sovereign Debt and Default,” Annual Reviews of
Economics, 5, 247–272.
53